Using different materials on each instance of an object with a mirror modifier
You will have to go around the mirror modifier and create 2 copies of the door each with different material, but with the same mesh they share between each other:


Get rid of the mirror modifier.
For each material slot set it to be linked to Object:


Make an instance of the door with Alt+D. This will duplicate the object, but link the same mesh between them. More on this can be found here:

What is a Duplivert, or a Dupliframe, internally?
Scale one object in the axis you want to -1 to mirror. Arrange them accordingly.
Assign different materials to the material slot(s).